Theorem eqvrelsymb 39399

Description: An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised and distinct variable conditions removed by Peter Mazsa, 2-Jun-2019.)

Hypothesis

Ref	Expression
eqvrelsymb.1	⊢ (𝜑 → EqvRel 𝑅)

Assertion

Ref	Expression
eqvrelsymb	⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))

Proof of Theorem eqvrelsymb

Step	Hyp	Ref	Expression
1		eqvrelsymb.1	. . . 4 ⊢ (𝜑 → EqvRel 𝑅)
2	1	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → EqvRel 𝑅)
3		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐴𝑅𝐵)
4	2, 3	eqvrelsym 39398	. 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵𝑅𝐴)
5	1	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → EqvRel 𝑅)
6		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝐵𝑅𝐴)
7	5, 6	eqvrelsym 39398	. 2 ⊢ ((𝜑 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐵)
8	4, 7	impbida 813	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   class class class wbr 5111   EqvRel weqvrel 38909

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-refrel 39301  df-symrel 39333  df-trrel 39367  df-eqvrel 39378

This theorem is used by:  eqvrelth  39404